The authors study the nonlinear heat equation \( u_t = u_{xx} + f(u),\) where a \(C^1\) function \(f(u)\) has two degenerate equilibria at \(\{0,1\}\) with
\[ f(0) = f(1) = f'(0) = f'(1) = 0, \]
but it is strictly positive for all \(u \in (0,1)\). Traveling wave solutions are defined by \(u(x,t) = u(x+ct)\) for a speed \(c \in \mathbb{R}\), where \(u(x)\) approaches \(0\) as \(x \to -\infty\) and \(1\) as \(x \to +\infty\).
The authors prove three important results for the existence, uniqueness, and stability of traveling wave solutions in this model.

1.

A unique traveling wave solution \(u^*(x)\) exists with the minimal speed \(c^* > 0\) such that \(u^*(x)\) is monotonically increasing in \(\mathbb{R}\) and \(u(x) \to 0\) as \(x \to -\infty\) exponentially fast. (The limit \(u^*(x) \to 1\) as \(x \to +\infty\) is always algebraic.)

2.

A unique traveling wave solution \(u(x)\) exists for any \(c > c^*\) such that \(u(x)\) approach both \(0\) and \(1\) algebraically slowly as \(x \to -\infty\) and \(x \to +\infty\) respectively.

3.

If the initial value is close to the limiting traveling wave solution \(u^*(x)\) with some exponentially small error, then the solution of the Cauchy problem is bounded from above and from below by the two limiting traveling wave solutions \(u^*_{\pm}(x)\) with the speeds \(c_{\pm}(t)\) that approach to \(c^*\) exponentially fast.

Appendix of the article gives an alternative existence proof that exploits ideas of the phase plane analysis and the qualitative theory of differential equations.